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Around 22 million account holders with the UK's biggest bank will be no better off as a result of the Bank of England's rate rise, despite Theresa May warning that benefits must be passed onto savers.

Yesterday the Bank of England hiked the base rate for the first time in ten years, up from 0.25 per cent to 0.5 per cent.

But last night Lloyds admitted that a rate rise was not on the cards for its 22 million current account holders, many of whom will use these accounts for building up cash savings.

This is because the accounts are not linked to bank rate, it said.

And Lloyds, along with Barclays and Halifax, have left tens of millions of savings account holders in the lurch by refusing to say whether they will receive higher interest rates.

All three claim their savings accounts are now "under review", with announcements expected in due course.

This is despite all three announcing that their tracker mortgage rates would be rising in line with the rate rise within hours of the Bank's announcement.

Last night Theresa May's official spokesman said she wanted to see banks pass on the interest rate hike to savers. He said: "Following the rate rise, we would expect to see higher interest rates to be passed on to savers".

"Some banks have already said they will increase rates on their savings products. We would expect others to follow suit."

Rachel Springall, money expert at savings website Moneyfacts, said savers and current account holders would be disappointed by some banks' apparent reluctance to increase interest rates.

She said: "A lot of people would expect that if there has been a Bank rate rise then they should receive more interest on their account. Whether their account is technically 'linked to base rate' makes no difference, if it's a variable rate people will want to feel the benefit and they will be disappointed if they get nothing."

Mark Carney, director general at the Bank of England, said: “We do expect it [base rate rise] to be passed on. Banks did pass on the cuts to their depositors, and we would expect competition to push it in the other direction [now].”